Symplectic topology and ideal-valued measures.
Joint with Yaniv Ganor, Leonid Polterovich and Frol Zapolsky.
Published in Selecta Mathematica.
Published Version (Preprint: arXiv:2107.10012)
We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.
Constraints on symplectic quasi-states.
Joint with Frol Zapolsky.
Published in International Journal of Mathematics.
Published Version (Preprint: arxiv:2407.08014)
We prove that given a closed connected symplectic manifold equipped with a Borel probability measure, an arbitrarily large portion of the measure can be covered by a symplectically embedded polydisk, generalizing a result of Schlenk. We apply this to constraints on symplectic quasi-states. Quasi-states are a certain class of not necessarily linear functionals on the algebra of continuous functions of a compact space. When the space is a symplectic manifold, a more restrictive subclass of symplectic quasi-states was introduced by Entov-Polterovich. We use our embedding result to prove that a certain `soft' construction of quasi-states, which is due to Aarnes, cannot yield nonlinear symplectic quasi-states in dimension at least four.
Approximation of quasi-states on manifolds.
Joint with Frol Zapolsky.
Published in Journal of Applied and Computational Topology.
Published Version (Preprint: arxiv:1812.10949)
Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. They were discovered as a part of an attempt to understand the axioms of quantum mechanics due to von Neumann. A very interesting and fundamental example is given by the so-called median quasi-state on the 2-sphere. In this paper we present an algorithm which numerically computes it to any specified accuracy. The error estimate of the algorithm crucially relies on metric continuity properties of a map, which constructs quasi-states from probability measures, with respect to appropriate Wasserstein metrics. We close with non-approximation results, particularly for symplectic quasi-states.
Symplectophagy.
Joint with Yaniv Ganor and Frol Zapolsky.
(Preprint: arXiv:2610.24862).
We introduce the notion of a symplectophagy, which is a symplectic isotopy satisfying conditions reminiscent of the symplectic camel. We present new symplectic rigidity results which take the form of obstructions to the existence of symplectophagies in certain situations. The main tool is the relative symplectic cohomology recently introduced by Varolgunes, as well as its extension to pairs, due to the authors and L. Polterovich, for which we establish an invariance property, which is the chief technical novelty of the present paper. An important role is also played by the recent calculations of relative symplectic cohomology in complex projective spaces due to Dickstein-Ganor.
Relative symplectic cohomology in complex projective spaces.
Joint with Yaniv Ganor.
(Preprint: arXiv:2606.17370).
Relative symplectic cohomology is an invariant of compact subsets of a closed symplectic manifold, introduced by Varolgunes. There are many examples of computations of this invariant over the Novikov field, but the collection of computed examples over the Novikov ring is still quite limited. One reason for this is that such computations require determining the relevant Floer complexes for Hamiltonians that are not necessarily C² -small Morse functions. In this work, we present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in ℂPⁿ. Our computation relies on explicit descriptions of Floer complexes, in the Morse--Bott setting with cascades, for J-shaped Hamiltonians on ℂPⁿ. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in ℂPⁿ.